Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response

In this paper, we studied a diffusive predator-prey system that incorporated three ecological features under homogeneous Neumann boundary conditions. These features included a cooperative hunting functional response, prey-taxis, and a time delay effect in the predator growth rate, all of which influenced predator-prey interactions. These three features, respectively, indicated that predators cooperated while hunting their prey, that predators tended to move in the direction of an increasing prey density gradient, and that a certain amount of time was required for the conversion of captured prey into predator growth. We first analyzed the occurrence of steady-state bifurcation by examining the role of the prey-taxis rate $\chi$. Furthermore, we examined the impact of the time delay effect on the occurrence of Hopf bifurcation, which involved the emergence of spatially homogeneous or nonhomogeneous periodic solutions, as well as stability switches at a positive constant steady state.